Lecture 35Plane surfaces.

Spherical mirrors.

Spherical ceiling mirror

Spherical make-up or shaving mirror

Image on a plane mirror

All the reflected rays seem to be coming from here!

Virtual image (the rays don’t come from it)

As explained in lecture 33:

How to find the image

• Draw two rays (one of them the normal to the surface, it’s a trivial one)

• Draw reflected rays.

• Extrapolate rays until they intersect.

ss’

s : location of the object

s’: location of the image

Positive in front of mirror

Negative behind mirror

Plane mirror:

s = -s’

Sign conventions

Position of object: s > 0 if object is on the same side of surface as incoming rays

Position of image: s’ > 0 if image is on the same side of surface as outgoing rays

Radius of curvature: R > 0 if center of curvature is on the same side of surface as outgoing rays

Magnification

y

Object’s height

Both y and y ’ can be negative if item is upside down!

Magnificationy

my

If m > 0, image is

upright.

If m < 0, image is inverted.

Example: For a plane mirror, m = +1

y ’

Image’s height

Reverse image

A B

C

A ’B’

C ’

Image on a plane mirror is virtual, upright and reversed.

Front and back are reversed

ACT: Refraction through a plane surface

object

s s’

Where does the spot appear to be?This image is:

A) Virtual

B) RealRays don’t really converge at that point, only their extrapolations.

The sign of s is: A) Positive B) Negative

Both object and incoming rays are left of surface.

The sign of s’ is: A) Positive B) Negative

Image is left of surface. Outgoing rays are right of the surface.

image

Image of an image (1)

Example: Two plane mirrors at 45°

1

2

Image of an image (2)

Use first image as an object for second mirror. Easier…

Image for mirror 1 / Object for mirror 2

Object for mirror 1Image for mirror 2

1

2

(if you look at mirror 2 first, you get another 2 images; and than there are also the images of the images of the images … see front page of lecture 33 notes)

Spherical mirrors

C = center of curvatureV = vertexR = radius of curvature

Optical axis

C

V

Robject

Concave R > 0

Optical axis

V

Robject

C

Convex R < 0

Concave spherical mirror

s’

sR

C

htan

tan

tan

hshshR

If α is small (paraxial approximation),

2

Thus,2h h h

s s R

1 1 2s s R

Independent of h valid for all rays (with small

α)

Focal distance, focal point

C

If object is very far, incoming rays are all parallel and s ∞

Rays converge at one point called the focal point F

F2R

f

f (focal length)

1 1 2f R

Focal length (spherical mirror)

1 1 1s s f

DEMO: Focal point

Getting the image

If a screen or a photographic plate is placed here, image will form on it

s > 0

s’ > 0Image is real and inverted.

F

1 ray parallel to the axis, reflection goes through F

1 ray to vertex, reflection at equal angle

Magnification (spherical mirror)

ss’

y

y ’

tany ys s

y sm

y s

sm

s

It looks a little odd: image seems to be in front of mirror (and we’re used to plane mirrors, where it’s behind)

DEMOS:

Mirror and bulb

Mirrors and pigs

ACT: Spherical mirror

Object

What do you see if you place your eye at the position shown?

DEMO:As you walk out look at the background with your face close to

the mirror.

Image

A. Smaller, inverted, real image of the arrow

B. Blurred image, hard to recognize

C. Virtual image of the arrowF

In-class example: Closer to the mirror

An object is placed at distance f/2 from this mirror. The image is:

s s’ < 0

F

A) Smaller, real and upright

B) Smaller, virtual and inverted

C) Larger, real and inverted

D) Larger, virtual and upright

E) None of the above

2 1 1

2f

sf s f

0, virtuals f 2s

ms

1 larger

0 upright

m

m

Convex spherical mirror

F

Everything is the almost the same for the convex mirror, except R < 0, so f = R/2 < 0.

Image is virtual, smaller and upright.

DEMO: Convex mirror

Appendix: Refraction through a plane surface (entire calculation)

object image

1 1 2 2sin sinn n

s s’

θ1

θ2

θ1θ2

s s’ 1 2tan tan 0s s s

n1n2

Where does the spot appear to be (for frontal view)?

2

2 21 1 2 1 2 2 2 2

02 22 2 1 2 1 11 22

21

tan sin cos sin 1 sin 1 sin

tan sin cos sin 1 sin1 sin

n nss n nn

n